The generator matrix

 1  0  0  0  0  0  0  0  1  1  1  0  1  0  1  X  0  X  1  1  X  1  1  1  X  1  0  1  1  0  1  X  0  0  1  X  1  0  0  1  1  0  1  1
 0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1 X+1  X  X  X  1  X  1  1  1  X  1 X+1  1  X  1  X  1  1  X
 0  0  1  0  0  0  0  0  0  0  0  0  0  X  1  1  1  1 X+1  1  X X+1 X+1  1  1  X  1  1  0  0 X+1  0  X  1  X  0  X  1  0  X X+1  X X+1  X
 0  0  0  1  0  0  0  0  0  1  1  X  0  1  X  0 X+1 X+1 X+1  1  1  1 X+1  1  0  X X+1 X+1  0  1  X  X  1  0 X+1  1  1  1  0 X+1 X+1  1 X+1  0
 0  0  0  0  1  0  0  0  1  0  X  1 X+1  0  0  X X+1 X+1 X+1  0 X+1  1  X X+1  1  1  X X+1  1  X  0  0  0  0 X+1 X+1  X  1  1  1  0  X  0  0
 0  0  0  0  0  1  0  0  1  X X+1  X X+1  1 X+1 X+1  X X+1 X+1  X X+1 X+1  0  0  X  X  1 X+1  X  X  X  0  X  X  0 X+1  0  1  1  X X+1 X+1  0  0
 0  0  0  0  0  0  1  0  1 X+1  0  1  X  1  1  X X+1  0  0  0  1 X+1 X+1  1  1 X+1 X+1  0  0 X+1 X+1  X  0  1 X+1  1  X X+1 X+1 X+1  1  0  X X+1
 0  0  0  0  0  0  0  1  X  1  X  1 X+1  1  1  1  1 X+1  1 X+1 X+1  0  X  1 X+1  X  X  0  0  0  1  1  X  1  1 X+1  X  0  1  1 X+1 X+1  X  0

generates a code of length 44 over Z2[X]/(X^2) who´s minimum homogenous weight is 31.

Homogenous weight enumerator: w(x)=1x^0+90x^31+248x^32+414x^33+668x^34+992x^35+1415x^36+1906x^37+2463x^38+3096x^39+3695x^40+4356x^41+4967x^42+5396x^43+5561x^44+5540x^45+5214x^46+4598x^47+3951x^48+3150x^49+2446x^50+1824x^51+1293x^52+898x^53+559x^54+356x^55+201x^56+112x^57+63x^58+28x^59+19x^60+8x^61+4x^62+4x^63

The gray image is a linear code over GF(2) with n=88, k=16 and d=31.
This code was found by Heurico 1.11 in 122 seconds.